# The Relativity of Indeterminacy

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## Abstract

**:**

## 1. Introduction

## 2. Relativity from Information Principles

**P1**—**Principle of relativity**: The laws of physics have the same form in every inertial frame of reference.**P2**—**Principle of finiteness of information density**: A finite volume of space can only contain a finite amount of information. (Note that assumption P2 is supported, in the context of general relativity, by Bekenstein’s bound [20], which intuitively states that since information is associated with a certain amount of energy, unbound densities of energy would degenerate into black holes).

## 3. Indeterminacy is Relative

#### 3.1. Locally and Independently Generated Randomness

- Local reality: Any two observers that locally overlap attribute the same truth values to empirical propositions (including the value “indeterminate”).
- Present reality: Any two distant observers at relative rest attribute the same truth values (including the value “indeterminate”) to empirical propositions about present events (i.e., lying on the same plane of simultaneity in their rest frame).

#### 3.2. “Classical” Correlated Randomness

#### 3.3. Quantum Correlated Randomness

## 4. The Block-Universe Picture(s)

## 5. Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Space-time diagram (in 1 + 1 dimensions) illustrating Rietdijk’s and Putnam’s argument for the alleged incompatibility between special relativity and indeterminism. The observers Alice (A) and Bob (B) are at rest in the blue reference frame, whereas Charlie (C) and Debbie (D), also at relative rest, move at a constant speed in the positive x direction (their transformed reference frame is depicted in red). The blue dotted line represents the plane of simultaneity in Alice’s and Bob’s rest frame at the instant in which Alice’s True Random Number Generator outputs the bit a (i.e., when it becomes determinate; event A). In the moving reference frame (red), however, when Charlie overlaps with Bob, he is simultaneous with Debbie, which in turn overlaps with Alice’s past (event A’) when a was not yet determinate.

**Figure 2.**Space-time diagram (in 1 + 1 dimensions) showing that for distant observers (in)determinacy is relative. Even if each of their local TRNG outputs a bit, becoming determinate from indeterminate, it is only in the overlap of their future light-cones that both bits become determinate. Note that both $a=0$ and $b=0$ are indeterminate in the entire white region.

**Figure 3.**Space-time diagram (in 1 + 1 dimensions) showing a relativistic scenario (readapted from Ref. [38]) in which different global quantum states are assigned in different regions of space-time.

**Figure 5.**Graphical representation of the “quantum” block-universe as envisioned by the Page–Wootters formalism and the many-worlds interpretation of quantum physics.

**Table 1.**Differences between epistemic (un)certainty and ontic (in)determinacy. The value of a variable a can be known only if it is determinate.

Epistemic | Known a | Unknown a | |
---|---|---|---|

Ontic | |||

Determinate a | ✓ | ✓ | |

Indeterminate a | ✗ | ✓ |

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Del Santo, F.; Gisin, N.
The Relativity of Indeterminacy. *Entropy* **2021**, *23*, 1326.
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**AMA Style**

Del Santo F, Gisin N.
The Relativity of Indeterminacy. *Entropy*. 2021; 23(10):1326.
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**Chicago/Turabian Style**

Del Santo, Flavio, and Nicolas Gisin.
2021. "The Relativity of Indeterminacy" *Entropy* 23, no. 10: 1326.
https://doi.org/10.3390/e23101326